Ramanujan Function

$\displaystyle \phi(a,n)$	$\textstyle \equiv$	$\displaystyle 1+2\sum_{k=1}^n {1\over (ak)^3-ak}$
$\displaystyle \phi(a)$	$\textstyle \equiv$	$\displaystyle \lim_{n\to\infty} \phi(a,n) = 1+2\sum_{k=1}^\infty {1\over (ak)^3-ak}.$

The values of $\phi(n)$ for

, 3, ... are

$\displaystyle \phi(2)$	$\textstyle =$	$\displaystyle 2\ln 2$
$\displaystyle \phi(3)$	$\textstyle =$	$\displaystyle \ln 3$
$\displaystyle \phi(4)$	$\textstyle =$	$\displaystyle {\textstyle{3\over 2}} \ln 2$
$\displaystyle \phi(6)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\ln 3+{\textstyle{2\over 3}}\ln 2.$